If \(x\) is small, so that \(x^2\) and higher powers can be neglected, then the approximate value for \(\frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}\) is

If the angle between the tangents drawn through the point \((-2,-1)\) to the parabola \(y^2=4 x\) is \(\theta\), then \(\tan 2 \theta=\)

If the lines represented by the equation $2x^2-pxy+2y^2=0$ are real then the value of ' $p$ ' lies in the interval

\([1+\sec 2 \theta][1+\sec 4 \theta]=\)

PQR is a right angled isosceles triangle with right angle at \(\mathrm{P}(2,1)\). If the equation of the line \(Q R\) is \(2 x+y=3\), then the equation representing the pair of lines \(P Q\) and \(P R\) is

Let $f\left(x\right)={\left(x+2\right)}^2-2, x\ge -2$. Then $f^{-1}\left(x\right)=$

If \(3 \cos x \neq 2 \sin x\), then the general solution of \(\sin ^2 x-\cos 2 x=2-\sin 2 x\) is

If 5 ampers of current is passed for 193 seconds through a solution containing Copper salt, \(0.32 \mathrm{~g}\) of copper is deposited. What is the oxidation state of the \(\mathrm{Cu}\) in the salt?

The solution of \(\left(1+y^2\right) d x-x y d y=0, y(1)=0\) represents a conic. Its eccentricity is

If the lines \(x+2 a y+a=0, x+3 b y+b=0\), \(x+4 c y+c=0\) are concurrent, then \(a, b\) and \(c\) are in

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

If \(x y \neq 0, x+y \neq 0\) and \(x^m y^n=(x+y)^{m+n}\), where \(m, n \notin N\), then \(\frac{d y}{d x}\) is equal to

The time taken for half of the initial amount of \(\mathrm{N}_2 \mathrm{O}_5\) to decompose is \(12 \mathrm{~min}\) at \(310 \mathrm{~K}\) and \(2 \mathrm{hrs}\) at \(300 \mathrm{~K}\). The activation energy of reaction in \(\mathrm{kJ}\) is \(\left(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)\)